Ann. Global Anal. Geon.

Vol. 3. No. 2 (1985), 197-217

SINGULARITIES ARE DETERMINED BY THE COHOMOLOGY

OF THEIR COTANGENT COMPLEXES

Bernd Martin

O. The cohomology of the cotangent complex characterizes

analytic singularities (resp. analytic mapping germs) up to iso-

morphism. The present paper generalizes a result by J.N.Mather

and S.S.T. Yau [6] that says that isolated hypersurface singu-

larities are isomorphic iff their moduli algebras are isomorphic.

This result turns out to be a special case of a general principle.

13] and [41 contain preliminary versions of the mentioned result.

where it was shown e.g. for the ,R-equivalence of functions

and for the 7(-equivalence of complete intersections that the

isomorphism type of the module of infinitesimal deformations of

first order T (Xo,o) determines the singularity (X ,o) up to iso-

morphism.

0.1. If the deformations of (Xo,o) are non-obstructed. then theabove statement is always correct. Otherwise, the higher co-

homology groups or all "infinitesimal families" of first order of

(Xo,o) have to be taken into consideration.

We proceed as follows: We choose n, p IN such that the singu-

larity (Xo,o) can be embedded into (An,o) and is defined by p

equations f J(n,p) :=m C X 1, ... Xn P. The set of all pos-sible representations of(X, o) in J(n,p) corresponds to an orbitof the contact group ((n,p) in J(n.p). In addition. we fix the

integers P2 .. Ps e IN such that the equations f have at most

Pi relations of (i-1)-st degree. Now the generating relations can

be represented by an element of a larger space J(n,p 1.. . p),

MARTIN

and all these representations form an orbit of a generalized

contact group '(n,p). To each of those elements of J(n,p) a

cotangent complex is assigned. The ation of 'X(n,p) on J(n,p)

induces isomorphisms of the assigned cotangent complexes and

their cohomology.

We have the "Mather-Yau equivalence":

If (n,p, .. . ,p5 ) is admissible with respect to (Xo,o),

then the following statements are equivalent:

(i) (Xo,o) is isomorphic to (X° ,o).

(ii) For i1 Ti(Xo,o) is isomorphic to Ti(X° ,o) as

{ X1... Xn} -module, where the isomorphism is in-duced by the action of K{(n,p).

(iii) T( 1 )(Xo,) is isomorphic to T (1)(Xo ,o) as

C (X 1 , X n } -module.

(Here T ) denotes the cohomology of the restricted cotangent

complex).

0.2. Similar statements hold both for singularities over a fixed

basis (S,o) (this generalizes the right-equivalence) and for

mapping germs (this generalizes the right-left-equivalence). The

proof involves transcendental methods. Simple examples show

(cf.[6]) that the Mather-Yau equivalence fails over non-algebraic-

ally closed fields or over fields of characteristic p O.

0.3. The concept of infinitesimally trivial families will be an

essential tool in proving the main theorem. We shall show that

these families are trivial. For hypersurface singularities this was

shown by K. Saito [8]. The statement is an analogue to the fact

that the triviality of the Kodaira-Spencer map implies the local

triviality of a deformation.

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MARTIN

1. THE GENERALIZED CONTACT GROUP

1.0. Let be the category of analytic 'germs of finite type over

C. A singularity over a base germ SE Ob f is a morphism

fo : Xo- S. A family of ' over T Obt is a pair consisting

of a T-morphism 'f: X > S x T and a fixed isomorphism

\,T/

X xT to} Xo . If X -- , T is flat, is called a deformation of o.

Now we describe the set of all possible coordinate representations

of '' and the associated syzygies as an orbit of a generalized

contact group. Together with S we .fix a minimal embedding:

S =- /Aq, OS= U1 ... Uq /'I(S) = ¢ u 1. .. . Uql I(S) r (U) 2

Let JT(n,p) := (X) OT X1 . . Xnp

1.1. A representative of y in JT(n,q+p) is a tuple

F = (H1 ... . Hq F1 ... ) F JT(n,p+q) , such that

x ' 0 T {XI /(F 1 .. .Fp) 0S x T X3 /(ul - H1 .... uq - Hq F)

and f (uj) = Hj mod (F),j · 1 ... q.

This implies:

G(U) e (S) - G(H) e (F)

(H,F)LT=t = (h,f) 6 J(n,q+p) represents ~o0

Remark: LYo has a representative in J(n,p+q) iff

n - emb dim (Xo ) = dim C(m OX / m 2 ) and. 0

p ~ i(X o ) := dimc(I(X)/m I(Xo)) for a minimal embedding

o= ¢ x/ I(X o) , I(Xo)~ m2

0

1.2. A resolution of the OS x Talgebra OX by free OS x T X} -

modules M sM . oS xI T (1)S ~ - 0 x(1

199

MARTIN

be representented as follows:

Let rk Mi p , (Po 1) {e i ) e() be a basis of Mi:Pi

I(e( jk e > 1 ) (la)i jk k

and, without loss of generality, we choose only representations

(1) where F(1) has the form:

Uj - Hj j s q

Fj.q otherwise (lb)

and F = (H,F) e JT(n,Pl) is a representative of ' .

Then (1) is completely described by

F(2) () p2po Psps-F=(F F ,.. .,F )) m XOs ST [ ...- "} 0 x T [ X } 5

=: JT(n,Pl ... p s ), m mT (X)

By construction we have:

(i) F( r ) F(r-1) = 0 , r = 2, ... ,s; F( ) is a Pr x Pr1 -matrix;

(ii) F(r) ,F(r) generate the (r-l)st syzygy module of F(1)

(iii) pr Pi

Remark: 1.) Let nr be the rank of the (r-l)-st syzygy module

of a minimal representative of 0o, then:

There is a representative of the described type for each resol-

ution (1) of OX in J(n,p 1 , . . ,ps ) iff n ~ emb dim X

P1 q + i(Xo) : nl and pi ni , (i 2)

Such a tuple is called (n,q,pl,...,ps ) is called admissible with

respect to L O n

2.) If r ( 1 ) for r 1 , then fo is calledr-1

non-obstructed. This is the case iff OX is a complete intersection0

200

MARTIN

and S is smooth.

1.3. Let two families YI and ' of tLf Over T, and two repre-

sentatives F' and F' of resolutions of the same type (n,q,p)

be given. A T-isomorphism 9 from QL to Lf':

_ j T- T T

can be lifted to a T-isomorphism y of T x An such that

' (F'1 . ' .) (F1 . Fs (2)and

' (Hj') = Hj mod (F 1 , ... Fs )

It is possible to extend Y to an isomorphism of the free reso-

lutions:

0 M M

A1I'

0 -. M -s.

We identify Ar

(aŽ) E

Ar(e( r ) )

and, because

G (q)p 1

.'' . M1 -- OS x T X} X

.. -. M1 ' OS x T X } °X

with the associated matrix

GI r(OS x T{X) Gpr

- a r)e( r ) , r = 2, ,s

of (2), A1 belongs to

- C GIp (OT X}).~~iTfX

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MARTIN

We obtain:

F - A 1 F,'( *)F(r) = A- 1 FI(r)( y)

--

(written as column vector)

Ar- 1 r 2

1.4. The generalized contact group ?J(T = 3JT(n,q,p 1,.. .p) is

defined as the semi-direct product of the groups

AutT(T x /An) x (q) x G x ... x GP1 P2 P

'(T is acting on JT(n,p) according to the formulas (3).

Remark: 1.3. implies that the set of all representations of re-

solutions of f of type (n,q,p 1, ... ps) is a XT-orbit in JT(n,p).of ~of ype n~qpl,..,p isa XT-obti JTnp)

Example: With one exception we consider only the "absolute"

contact group XK = to . For s = 1 JX (n,q,p) contains a

lot of known groups:

(i) X. (n,q,q) = R(n) is the group of right-equivalences on J

(ii) X(n,o,p) is a subgroup of {(n,p), the contact group

in the sense of J.N. Mather 51, but the orbits of both

groups coincide in J(n,p).

(iii) Let S = C1 , q = 1: X (n,l,p) defines a generalized right-

equivalence of functions on singularities. For hypersurfaces

this equivalence was introduced by A. Dimca in 11, where

the simple .(n,1,2)-orbits have been classified as well.

2. INFINITESIMAL DEFORMATIONS

Infinitesimal deformations of first order are classified by the first

cohomology group of the cotangent complex. Now we give a short

description of Palamodov's construction of the Tjurina-resolvent

71]:

(3)

(n, q).

202

MARTIN

2.1. The Tjurina-resolverit:

Let 0:; -: O be a local homomorphism of analytic algebras.o O 0 S ~d

(R,I) is called a Tjurina-resolvent of o if the following con-

ditions are satisfied:

(i) R = R F v 1 ..... Vnl is a free graded anticommutative

R -algebra, deg v. = Iv.l 0 , v * vj = (-1)liVjlv.v.

(ii) R °= O X 1..... Xn} is a free OS-algebra and

Ik 0..Rk ' R. R+o ° X

OsOS

is a resolution of OX by free R-modules.o

(iii) I : R - R is a derivation of degree 1, i.e.

I(a b) = I(a)b + (-1)lalal(b) .

The Tjurina-resolvent is uniquely determined up to homotopy.

R has at least nr - ( r-1) generators of degree -r, r > 1, and

n 1 generators of degree -1, respectively, (cf. 1 .2.). To each

representative F' of a resolution (1) of 'o a Tjurina-resovent

is assigned:

R := OS X. X, ' v N , t PiPi- 1

.... nj 1'***' N i=1

The v.'s are associated with the generators of M and get the

degree -r, r = 1 ... s. Due to (la) and (lb) I is defined on

the generators by the corresponding map I of the e(r). The re-

solvent (R,I) constructed in this way is not minimal.

Example: Let : C U1 j ... 'Uq} 0X

OX C {X 1... Xn} / (f...f) be a complete inte intersection

n = emb dim XO p = i(X o ) . Then 0oO is non-obstructed and aminimal resolvent is obtained by:

R C U,X} F v,....vp+q 1 deg =. -1

203

( Ui - h , i 1,... ,q; hia representative of (Ui )

I(v.) = f; 'f SjV 4 . otherwise

2.2. The cotangent complex

Let : = Der 0 (R,R); is a graded OS-module and

d' be Os-linear, d(R I) C RI+k

Rk := t: -- R R (ab) = (a)b + (-1) kl al aS(b)

Every 6e Rk is uniquely determined by the images of the vi,

Civil = d = -k, (vi ) Rd+k and by lRO ri Rk

for k - O.

Therefore, Rk is a free R-module:

Rk = Rk + Lo R +k and n(o) = # vilvi

By t ,O' ] := Sc' - (-1) 16ll6' l &&' R is a graded Lie-algebra

and a cotangent complex by d(6) C ,l 1 = 61 - (-1)1l 16

The cohomology of the cotangent complex T ker d /lm d. 1 is

independent of the choice of a resolvent and has the structure

of an R-module. By direct calculations we obtain:

(i) Ti( o) = 0 for i O ,

(ii) To( O) = Deros(Ox ) ,

(iii) If Lo is non-obstructed, then Ti( 0o ) = O for i> 1

(cf. the following example),

(iv) T1( .f) classifies the infinitesimal deformations of first

order of iP: up to isomorphy: [ ¢] Q6T1 is uniquely deter-

mined byh. i q for Ivil = -1

d (v.) = f m =-f iq , otherwise

then ~c is represnted by (h + E h, f + T) EJC cFI(n p

204 MARTIN

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2.3. Example (continued):

Ri 0 for i b 2 , then C- R.i is determined by

and thus Ti = 0 for i b 2 and ker d= R1

Let E Ro be given by & IR = and0 i

gi,g9ij E Ro, then d(c) is determined by d(v i) :

d '(vi) (fi ( 1 ) ) - I( gijvj )

A j (1)

k aXk- g9jf(1)

CF(vi)e- i_1 a ,

6(vi) Zg ij

f(1) (f(1) This means: Im d : Ro a f(l) ((l))i 1 = R

and thus

T1 ( pO) = C {[X p+q, t(h,f) = *X q/ T(h,f)0

t(h,f) = T C X) af + (f) C x 3 P'q

t(h,f) = t(h.f) OxP q0

(4)

2.4. The restricted cotangent complex

If we consider only the syzygies up to order k, we obtain the

k-th restricted resolvent:

R(k) := R R_1 ... Rk =k R/I k . Let R(k) be the k-th asso-

ciated cotangent complex and let Tik ) be the k-th restrictedcohomology. By construction: Tr k) 0 for r k and

Tk = - (k) =(k) = Rk/ Im dk-l1 Ik k Similar to example 2.3 we have:

T(1)( o) = 0 X / t(h,f) if f = (h,f) eJ(n,pl) represents Lo.

The 0O-module structure of T 1 ) is given by U ~ := h. 5, ~ e T(l)

Remark:

- T(11) classifies all infinitesimal families of yo of first order up

to isomorphy.

T1 is an OS-submodule of T(11 cf. 2.2 (iv).S O~~~~~1

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MARTIN

2.5. The action of the generalized contact group

Let (n,q,p 1,...,p s) be admissible with respect to S, then

T'( o) has a natural Os[X} -module structure.

Let f = (f ,f(2,... f(S)) be a representative of given type of

a resolution of Cyo. Any other representative f has the form

k f , kE '(n,q,p1,...,ps). k induces an isomorphism r

of the associated resolvents R R', which is compatible with I:

k = (,A 1.... As)

:R - 05sX3 r v ... VN1 - R' = 05s X r V 1... N v

X i : P Y *(Xi

VI

and induces an isomorphism i of the cotangent complex

because is compatible with I.

: R , (): O -1

Analogously, k(r) = (Y ,Ai. Ar) E (n,q,p 1 ... .r ) induces

an isomorphism of the restricted cotangent complex.

Restricted to elements of degree i i is an OS{X -module iso-

morphism and we get OStX } -module isomorphisms

T'( fo) o T( y ), o is represented by f' .

3. INFINITESIMALLY TRIVIAL FAMILIES

The notion of infinitesimally trivial families is an effective tool

to prove the triviality of a family. Roughly spoken, a family

is called infinitesimally trivial if the lifting of a family to each

tangent direction of the basis T is trivial, i.e. isomorphic to

fx idl , I = Spec C I , £ 2 = . If T is smooth, then every

infinitesimally trivial family is trivial, i.e. M 't o x idT.

In other words, we obtain a local version of the theorem on

the Kodaira-Spencer map associated to a deformation of a com-

plex space: A deformation is trivial iff the Kodaira-Spencer map

vanishes .

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MARTIN

3.1. The Kodalra-Spencer map

The Kodaira-Spencer map of a deformation t : X - S x T of

a singularity '9 o is a map c, .: Der OT . T( 1)( 9) defined as

follows:

For every derivation Isa of OT there is a morphism ta :I- Twhich can be lifted to a T-morphism

prt : T x I- ~ T x t (I) ; T x T p 1 T.a a

assigns to ca the class in T(1)() of the family ofa (1)(~) of the family of ()

over I fa := XT ta induced from by T

Definition: A family of f0 over T is called infinitesimally

trivial if the Kodeira-Spencer map L{f vanishes identically or, in

other word, if for every tangent vector a of T the family a

induced by If over I is trivial.

Example: Let T be smooth, 0 T = C T 1T .. Tr} , t (Ti ) = ai,

ai. C be a tangent vector of T, then t*(h(T)) = h + a T.

is the restricted Taylor expansion of h in direction a. Let X be0

an arbitrary singularity, OX = C {X 1 ,...Xn} / (fl, .....,) and

let Y: X - T = be a family of X, OX = CT.X / (F)

Then L? is infinitesimally trivial iff ta (F) = F + E. L ai AF

is a trivial deformation of F iff aF e t(F) C TX} P

This example can be generalized:

rLemma: Let be a family of 'f0 over T = ,r and let

"F = F + F'* be a representative of Fa, F = (H,F) be a

representative of f . Then Ya is trivial iff F' 6 t(H,F). (5)

Proof: Let F' E t(H,F) have the form

F'= a; - F + (bkj) F , a , bkj6EC T,X , then (Y,A)

determines an element k from f[(r+n,q,p+q) , where

: X; X. + a., Tj - T. and A=E (O I bk )

207

208 MARTIN

It holds that kF = F*.

On the other hand, let F = (H,F) J(r+n,q,p+q) be an arbitrary

representative of Yq and let *Fe Jl(r+n,q,p+q) be a representa-

tive of Ya . If %fa is trivial, then there exists

k = ( ,A) e XI(r-n,q,p+q) such that kF = F . Without loss of

generality, let k F0 = 1 . Then k is given by

) : XiA- X + &a., A = Ep+q + (O I bkj and

kF* F' + ( a - F + (bkj)F) = F

Comparing the coefficients of the last equation we obtain (5).

3.2. The following proposition is a generalization of Lemma 3.5

of 8] :

Proposition: Let 'q be a family of o over T, T be smooth, then

q is trivial iff is infinitesimally trivial iff the Kodaira-Spencer

map LIp vanishes identically.

Proof: Let F be a representative of f . If o is infinitesimally

trivial, then, by Lemma 3.1:

a Fe t(H,F) for i = 1, ... r, r dim T.

Induction on r:

r = 1 : Let F + ai a F = (b k(T,X))Fk i i k=1

The vector field + E a5(XT) has an integral curve

through (T,X) given by (V . .. V)e C T,X,YJ n1

av.Oy = a(V) Vi(T,X,O)= X, i = 1, . . ,n

av.al- = 1 Vo(T,X,O) = T

Then F*;(V), ... Fp+q(V) are the solutions of the differential

equation

MARTIN 209

az.Y k b jk(V)Zk, j 1,...,paq (6)

for the initial value Z(T,X,O) = F.

We give the following formulation:

F'(V) = (S )F j = 1 .... p+q

k = 1, ... ,p

and

S jk t I'J-qk j>qjk I Y=O 0O otherwise

Inserting this into (6) we obtain:

as F* 1 ...,p+q (7)

F (bjk(V))(Sk l,k = 1.... p

Hence, we get the solution of (7) by the differential equation

j.L = Z b jk(V)W kl with the initial valuea)y k

Wl °= 'j-ql j>q

y=O otherwise

Let Y* be defined by X: - V(O,X,T) and

A q | Sjk(O,X,T , then k = (Y,A)e '(T(n,q,p+q) and

k (H,F) = (h,f) , (h,f) = (H,F) I represents f0 , henceT=O

0oX idT

Induction step:

Let T' = Spec C tT .Tr1 . . Spec C T

T = T' x T , F' = F* I F' F by the basis of the

r

induction and F"* f* by the induction hypothesis.q.e.d.

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Corollary: Let : S x be a family of S-germs over a

smooth connected curve T . If the germ of Up is infinitesimally

trivial at any point t e T of the curye, then () is a T-trivial

family.

Proof: The trivialization of each germ )t has a representative

defined in the neighbourhood U(t) T of t . Pasting together the

local trivialization then yields a global one.

4. THE GENERALIZED MATHER-YAU EQUIVALENCE

Proposition: Let f0 : XO - S be a singularity and

(n,q,p 1 ... ,ps) be admissible, then the following statements

are equivalent:

(i) ~00 and 'o are isomorphic in S

(ii) For all r r: Tr( Po) Tr( qt') as OS X, ... Xn

module, where the isomorphism is induced by

X(n,q 'Pl ... Pr)

(i) T 1 )( T o ) - T1)( o ) as OS X} -module, where the

isomorphism is induced by 7(n,q,p 1)

4.1. Proof:

(i)-(ii) by 1.3 and 2.5.

(ii)- (iii) by 2.3.

(iii) . (i) We choose representatives f and f', respectively, in

J(n,pl ) .

The isomorphism in (iii) is induced by some kW(n,q,p1 ). We re-

place f by k f , then t(h,f) = t(h',f').

FS := (1-T)f + Tf'* represents a family of S-germs over A1,

such that the germ 95t of at t = 0 (resp. t=l1) defines a family

of LFo (resp. Ly' O )

210

Lemma: Up to a finite number of values t t, . . .t we have

t(ht,ft) = t(h,f).

Proof: We have t(ht,ft) t(h,f) = t(h',f'). t(ht,f t ) is a finitely

generated submodule of C X P with a system of generators

ml(t), . . . ,mM(t) and mj(t) = Cjk(t)mk(O). Up to a finite number

of values that are the zeros of det cjk(T) 6 ¢ T cjk(t) is aregular matrix.

Let U = C - {t 1 .... tt Q , at every point of U the germ (Ptis infinitesimally trivial, because

Ft = f - f 6e t(h,f) t(Ht,Ft) . This statement is obvious

for the last p components Ft of F . As the OS-module structures

of T1) ( 0 ) and T )(;) coincide, we have (h'j - hj)ek E t(h,f).

Now U is connected and by the corollary U is a trivial family

over U, hence, F° TO and F; ITO represent isomorphic

S-germs.

4.2. Interpretation of special cases

Due to the proposition it is sufficient to know the cohomology

T' (f0) to determine the singularity up to isomorphism. But, it is

difficult to decide whether an isomorphism of T' is induced by the

contact group or not. If we consider non-obstructed singularities,

then the statements (ii) and (iii) coincide (cf. 2.3):

(a) Let S = 0} and X be a complete intersection: Any C XJ -

module isomorphism of T1 ( O) has a lift to C {X} pl (cf. (4)).Any such isomorphism is given by an isomorphism Y' of ¢ X}

and a matrix A GLp (C X ) , i.e. induced by the contact

group 5{(n,O,p 1 ). Hence, we obtain a generalization of theMather-Yau result for isolated hypersurface singularities to the

case of complete intersections with arbitrary singularity.(b) Let S = Aq, o be smooth and let X = An",o be smooth, then an

1 0isomorphism of T ( o ) is induced by (n,q,q) = R(n) iff it is

induced by a ring-isomorphism Y* of : [IX . This corresponds

MARTIN 211

to the right-equivalence of q-tuples of functions on C ,o.

For q 1 the Mather-Yau equivalence implies:

Two functions f,f': Cn,o - C1 ,o are right-equivalent iff the

local algebras Q f } / ( . .. . ) and Qf are

isomorphic as CTI-algebras.

For functions with isolated critical points at 0 this was shown

by J. Scherk 9], who used the methods of [6].

(c) Let S be smooth and X0 be a complete intersection: The

isomorphy of such singularities corresponds to a "right-

equivalence" of q-tuples of functions on complete intersections.

As in (a) an isomorphism of T lifts to an isomorphism of

CX} p 1 given by Y and A. A sufficient condition, that ( ,A)

on T1 induces an O05 X]-isomorphism, is that A is an element

of p (q), i.e. is induced from an element of XJ(n,q,p 1 ).p1

For obstructed singularities we conclude the following:

(d) Let S = t 0 , X be arbitrary:

Tr( T0) is a submodule of Rr/ Im drl, Rr a free C[Xi-module.

An isomorphism Tr( o) Tr( f0) is induced by ( if it can

be extended to a C X}-module isomorphism of Rr . Analogous

to (a) each of these isomorphisms is admissible.

(e) In the general case Tr( tfo ) , (r , 1), is a submodule of

Rr / Im dr-1l Rr a free 05stX-module. An isomorphism is in-duced from the contact group if it can be extended to an

Os5 X}-isomorphism of Rr. For T C T(1) - C[XjP 1/ t(h,f) wehave in addition (cf. (c)):

An isomorphism of T(11) is admissible if it has a lift to C[X}p ,

given by ( y,A) and A T, (q). It is still an open question

whether the condition A (q) is really necessary, i.e.p1

whether any OX}I-module isomorphism of T( ) is induced

from X .

212 MARTIN

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5. THE MATHER-YAU EQUIVALENCE FOR MAPPING GERMS

For mapping germs and their deformations the present theory canbe established in a rather analogous way. The main points willshortly be described here. Thereby we shall, ignore the highercohomology groups and equivalence will be given only for therestricted cohomology T 1) The theory of the cotangent com-plex of a morphism between analytic spaces has been developpedby H. Flenner [2].

5.1. The group of the generalized right-left equivalences

In contrast to families of singularities L o over S, where thebase germ S is fixed, we now also deform the image germ. Bya family ?Z of a mapping germ l o : XO ~ SO 6& over T wemean a T-morphism X : S with an isomorphism

A family 7 can be represented by

F' = (HF,G) EJT :=JT(n,p+q) x JT(q,r) , such that

OX - OT I X 1 ... x n} / (F1 .... Fp

OS = OT tU 1 ... Uq / (G1, ... Gr)

(Uj)= Hj mod (F) j = 1 .. q

A tuple (n,q,p,r) is called admissible with respect to o if

7o has a representative in J.This is the case iff

n ' emb dim X,0

q ' emb dim S,

p i(Xo ) and r i(S ) .

The set of all representatives of fixed type forms an orbit of a

group T(n,p,q,r) in JT-. T is the semidirect product of

'7T(n,q,p+q) and T(q,Or).

Jr T acts on JT as follows: Let a L91T be represented by

,( 1 ' ) E T(n,q,p+q) and ( t2' A3 )E T(q,O,r);2

213

then a F =F * and

F : A 1 ; (F)

G := A31 (G)

H = Y; (H '¶- ) - A1A 1 Y; (F)

If T = 0O and p = r = 0 (i.e. X and S are smooth), then

A[ = - (n) x t (n) and the action of on J(n,q) is just the

right-left-equivalence in the sense of Mather ([51).

5.2. Infinitesimal families

Infinitesimal families 7 of 0o on I = spec QC F] have representa-

tives of the form f* + f* , f = (h',f' ,g) e. C [x P+q e C U r

A family 7' is trivial iff' 'e t(h,f,g). t(h,f,g) is a submodule of

C X} qP P® : tU3 r, which is composed of

t = t(h,f) = C LX} a f ) + C [X3 P'q f , a : tX -sub-

module of t X P+q ,

t 2 := ' U} r a tUl-submodule of : U) r

and

t3 ={ Ai f + A (h) A = (A1 ...., Aq, 0 )t, Ac t

a C {U} -submodule of T {X} q(D C: U r

T() ( O) C (x j}Pq (C r / t(h,f,9g) classifies all infinitesi-

mal families of first order of 1 up to homotopy.

The structure of T1 is richer than that of a C u3 -module, due

to the subdivision into the three components. The action of . on

IJ induces C {Ul -module isomorphisms of T( 1 )( 7Zo) , which pre-

serves this finer structure.

5.3. Infinitesimal triviality

The notion of infinitesimal triviality (cf. Definition' 3.1) can also be

applied to families of mapping germs and we get the same proposi-

tion.

214 MARTIN

MARTIN

Proposition: Let T be smooth and let 72 be a family of 7 0 over

T. 7 is infinitesimally trivial iff is trivial.

Proof: The proof is almost the same as in 3.2. Without loss of

generality let dim T = 1 and let F be a representative of ? .

The infinitesimal triviality of 7 implies:

-- F· . ii=1ai(T,x) -- F' + C( F = BF* + c(T,P4)

where B is a (q+p+r ,q+p+r)-matrix of the form

B1(T,X)

B2 (T ,X)

0

O

0

B3 (T,U) )and c =(c 1 ... Icq,O,... ,0).

We obtain integral curves

V = T + Y, V* .. ,Vn of + - a a

and VoVn * n+q of a + j aj

Vi C tT,X,Y}

Vn+j C T,U,Y .

Then F(V) is the solution of the differential equation

az B(V)- Z c(Z).

Here formulation (7) has the form:

F*(V) - ( H ) - c(H) = S F ,

SS = 0

S1 0

S2 00 S3 )

and S fulfills the differential equation

Hence we obtain a e 4T by

; : Xi ;-- V(O,X,T)

); : Uj o> Vn+j(O,UT)

bw B(V) w.Ty- B(V) W.

(0

0

0

215

MARTIN

i :i I T : O, Y := T

and then VJ; F= Af

t; G = A3 9

H In ; Alf h- Y+ A 1 A2 Y 1 F

hence,

aF = f*q.e.d.

5.4. Proposition: Let 7 X - S be a mapping germ and let

(n,p,q,r) be admissible with respect to 7 o. Then the following

statements are equivalent:

(i) r7 o is isomorphic to o

(i) T(1)( 0) T(1) as C: U } -modules induced by an ele-

ment of ,'(n,p,q,r).

The proof is a repetition of the arguments of 4.1.

Remark: If X and S are smooth, then an isomorphism0 0

T(1)( 'o) = T(1)( 7 is admissible iff it is induced y a pair of

ring-isornorphisms of C Xj and C U)

REFERENCES

[1] Dimca, A., Function germs defined on isolated hypersurfacesingularities, Prepr. 4/83, Increst, Bucaresti.

[21 Flenner, H., Uber Deformationen holomorpher Abbildungen,Osnarbrucker Schriften zur Mathematik, Heft 8, 1979.

[3] Martin, B., A generalized Mather-Yau Equivalence, Prepr.Nr. 60, HUB, 1983.

[4] Martin, B., Singularitaten sind bestimmt durch ihre infinitesi-malen Familien, Prepr. Nr. 75, HUB, 1984.

216

MARTIN

[5] Mather, J.N., tability of C-mappings, III, IHES, Publ.Math. 35(1969), p. 127-156.

[6] Mather, J.N., Yau, S.S.T., Classification of isolated hyper-surface singularities by their moduli algebra,Invent. Math. 69(1982), p. 243-251.

[(7 Palamodov, V.P., Deformations of complex spaces (Russian),Uspech. mat. nauk, 31:3(1976), p.129-134.

t[8 Saito, K., Theory of logarithmic differential forms and loga-rithmic vector fields, J. Fac. Sci. Uni. Tokyo,27:2(1981), p. 265-291.

t9] Scherk, J., A propos d'un theoreme de Mather et Yau,C.R. Acad. Sc., Paris, t. 296, ser. I, No. 12,1983.

Bernd Martin, Humboldt-Universitat zu Berlin, Sektion Mathematik,

DDR-1086 Berlin, PF 1297, German Democratic Republic

(Received November 10, 1984)

217